Let $E$ be a Euclidean space of dimension $N$. We denote by $(|)$ the inner product and $\|\cdot\|$ the associated Euclidean norm. Let $u$ be a self-adjoint endomorphism of $E$. We define $q_u : E \rightarrow \mathbf{R}$ by $q_u : x \mapsto (u(x) \mid x)$ and we assume that for all $x \in E$, $q_u(x) \geq 0$. State the spectral theorem for the endomorphism $u$. What can be said about the eigenvalues of $u$?

Consider $a = (a_1, \ldots, a_d) \in \left(\mathbb{R}_{+}^{*}\right)^d$ and $b = (b_1, \ldots, b_{d-1}) \in \left(\mathbb{R}_{+}^{*}\right)^{d-1}$ and introduce the matrix $$M = \begin{pmatrix} a_1 & b_1 & 0 & \ldots & 0 & 0 \\ a_2 & 0 & b_2 & \ldots & 0 & 0 \\ a_3 & 0 & 0 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{d-1} & 0 & 0 & \ldots & 0 & b_{d-1} \\ a_d & 0 & 0 & \ldots & 0 & 0 \end{pmatrix}.$$
(a) Justify that there exists a unique pair $(\lambda, \pi) \in \mathbb{R}_{+}^{*} \times \mathscr{P}$ such that $\pi M = \lambda \pi$. Express $\pi$ explicitly in terms of $a$, $b$ and $\lambda$.
(b) Show that there exists a unique $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ such that $\langle \pi, h \rangle = 1$ and $$Mh = \lambda h.$$
(c) Deduce that the sequence $\left(\lambda^{-n} M^n\right)_{n \geqslant 1}$ converges as $n$ tends to infinity and give an expression for its limit in terms of $h$ and $\mu$.

Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Show that $D$ is symmetric, with non-negative coefficients and zero diagonal, and has eigenvalues $\lambda_1, \ldots, \lambda_n$, with $\lambda_1$ having eigenspace of dimension 1.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Show that the roots of $\varphi_A$ in $\mathbb{C}$ are exactly the eigenvalues of $A$.

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Determine the eigenvalues of the matrix $A$.

Problem 2

Answer the following questions about the square matrix $A$ of order 3:
$$A = \left( \begin{array} { c c c } 3 & 0 & 1 \\ - 1 & 2 & - 1 \\ - 2 & - 2 & 1 \end{array} \right)$$
I. Find all eigenvalues of $A$. II. Find the matrix $A ^ { n }$, where $n$ is a natural number. III. The square matrix $\boldsymbol { B }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { B } = \boldsymbol { B } \boldsymbol { A }$. Prove that any eigenvector $\boldsymbol { p }$ of $\boldsymbol { A }$ is also an eigenvector of $\boldsymbol { B }$. IV. Find the square matrix $\boldsymbol { B }$ of order 3 that meets $\boldsymbol { B } ^ { 2 } = \boldsymbol { A }$, where $\boldsymbol { B }$ is diagonalizable and all eigenvalues of $\boldsymbol { B }$ are positive. V. The square matrix $\boldsymbol { X }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { X } = \boldsymbol { X } \boldsymbol { A }$. When $\operatorname { tr } ( \boldsymbol { A } \boldsymbol { X } ) = d$, find the maximum of $\operatorname { det } ( \boldsymbol { A } \boldsymbol { X } )$ as a function of $d$.
Here, $d$ is positive real and all eigenvalues of $X$ are positive. In addition, $\operatorname { tr } ( M )$ is the trace (the sum of the main diagonal elements) of the square matrix $\boldsymbol { M }$, and $\operatorname { det } ( \boldsymbol { M } )$ is the determinant of the matrix $\boldsymbol { M }$.

Let m be a positive real number and $u = \left[ \begin{array} { l l } x & y \end{array} \right]$. Given that
$$\mathrm { u } \cdot \left[ \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right] = \mathrm { u } \cdot \left[ \begin{array} { c c } \mathrm { m } & 0 \\ 0 & \mathrm {~m} \end{array} \right]$$
the matrix equation has infinitely many solutions for u, what is m?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) 3
E) 4